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 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 A. Example: exponential of stochastic process.
 B. Example: integral of t_dW.
 C. Example: integral of W_dW.
 D. Example: integral of W_dt.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Example: integral of t_dW.

he goal of this section is to calculate distribution of . By the property ( Ito isometry ) we have The integral is a limit of a sum of normal variables. Each of the variables has zero mean. Hence, the result is a normal variable with zero mean. We conclude,

 (Int t dW)
for some standard normal variable .

 Notation. Index. Contents.